An efficient branch-and-cut algorithm for the multiple probabilistic covering location problem
Yan-Ru Wang, Wei-Kun Chen, Ivana Ljubić
TL;DR
The paper tackles MPCLP, which seeks to open $K$ facilities (allowing co-location) to maximize probabilistic, jointly covered demand under a flexible joint coverage model $p_j( heta, S) = heta ( ext{max}) + (1- heta)(1- ext{prod})$. It introduces a compact MINLP reformulation whose variable count grows linearly with the problem size and develops an LP-based branch-and-cut algorithm that uses submodular and outer-approximation inequalities to linearize nonlinearities, with node-wise separation. Two families of strong valid inequalities—enhanced outer-approximation and lifted subadditive—are derived to tighten the LP relaxation, and their separation is integrated into the B&C to accelerate convergence. Computational experiments on 240 MPCLP instances show that the proposed method significantly outperforms the Alvarez-Miranda2019 B&C, solving 205 instances to optimality (vs 153) and solving 57 previously unsolved instances within one hour, underscoring the approach's scalability and practical impact. Overall, the work delivers a compact, scalable framework for MPCLP with substantial gains in solution speed and breadth of solvable instances under realistic probabilistic coverage models.
Abstract
In this paper, we consider the multiple probabilistic covering location problem (MPCLP), which attempts to open a fixed number of facilities to maximize the total covered customer demand under a joint probabilistic coverage setting. We present a new mixed integer nonlinear programming (MINLP) formulation, and develop an efficient linear programming (LP) based branch-and-cut (B&C) algorithm where submodular and outer-approximation inequalities are used to replace the nonlinear constraints and are separated at the nodes of the search tree. One key advantage of the proposed B&C algorithm is that the number of variables in the underlying formulation grows only linearly with the number of customers and facility locations and is one-order of magnitude smaller than that in the underlying formulation of a state-of-the-art B&C algorithm in the literature. Moreover, we propose two new families of strong valid inequalities, called enhanced outer-approximation and lifted subadditive inequalities, to strengthen the LP relaxation and speed up the convergence of the proposed B&C algorithm. In extensive computational experiments on a testbed of 240 benchmark MPCLP instances, we show that, thanks to the small problem size and the strong LP relaxation of the underlying formulation, the proposed B&C algorithm significantly outperforms a state-of-the-art B&C algorithm in terms of running time, number of nodes in the search tree, and number of solved instances. In particular, using the proposed B&C algorithm, we are able to provide optimal solutions for 57 previously unsolved benchmark instances within a time limit of one hour.